Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T06:25:34.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control

Published online by Cambridge University Press:  28 May 2015

Hans-Görg Roos  and
Christian Reibiger
Hans-Görg Roos*
Affiliation:
Technische Universität Dresden, Institut für Numerische Mathematik, 01062 Dresden, Germany
Christian Reibiger*
Affiliation:
Technische Universität Dresden, Institut für Numerische Mathematik, 01062 Dresden, Germany
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection-diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.

Keywords

34E0534E2065L1065L1165L6065L70Convection-diffusionlinear finite elementsa priori analysislayer-adapted meshessingular perturbedoptimal control
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 562 - 575
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Becker, R. and Vexler, B., Optimal Control of the Convection-Diffusion Equation Using Stabilized Finite Element Methods, Numerische Mathematik, 106 (2007), pp. 349–367.CrossRefGoogle Scholar
[2]Cen, Z., Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations, International Journal of Computer Mathematics, 82 (2005), pp. 177–192.CrossRefGoogle Scholar
[3]Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2001.CrossRefGoogle Scholar
[4]Hinze, M., Yan, N. and Zhou, Z., Variational Discretization for Optimal Control Governed by Convection Dominated Diffusion Equations, Journal of Computational Mathematics, 27 (2009), pp. 237–253.Google Scholar
[5]Linss, T.Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations, Computing, 79 (2007), pp. 23–32.CrossRefGoogle Scholar
[6]Linss, T.Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Springer-Verlag, 2010.CrossRefGoogle Scholar
[7]Linss, T. and Stynes, M., Numerical Solution of Systems of Singularly Perturbed Differential Equations, Computational Methods in Applied Mathematics, 9 (2009), pp. 165–191.CrossRefGoogle Scholar
[8]Lube, G. and Tews, B., Optimal Control of Singularly Perturbed Advection-Diffusion-Reaction Problems, Mathematical Models and Methods in Applied Sciences, 20 (2010), pp. 375–395.CrossRefGoogle Scholar
[9]Roos, H.-G., Stynes, M. and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd Edition, Springer-Verlag, 2008.Google Scholar
[10]Tröltzsch, F., Optimal Control of Partial Differential Equations, American Mathematical Society, 2010.Google Scholar